#449731

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,4)(10,13)(14,16)(17,18,22,23,21,25,19,26)(20,28,27,32,30,31,29,24);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,28,32,31);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(10,13)(14,16)(17,20,21,30)(18,24,25,32)(19,27,22,29)(23,31,26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356927);